If Bohr's model is applicable to an atom `._100X^256` then its orbital radius will be

Options

A. `5.29xx10^(-11)` m
B. `1.32xx10^(-11)` m
C. `2.64xx10^(-11)` m
D. `1.76xx10^(-11)` m

Explanation

To find the orbital radius of the atom denoted as `._100X^256` using Bohr's model, we will follow these steps:

### Step 1: Identify the Atomic Number and Mass Number
The atomic number (Z) of the atom is 100, which indicates that there are 100 protons and, in a neutral atom, also 100 electrons. The mass number is 256, but for calculating the orbital radius, we only need the atomic number.

### Step 2: Determine the Orbital Number (n)
In this case, we need to find out which is the outermost orbit that contains electrons. The total number of electrons is 100. We will fill the electrons in the orbits according to the maximum capacity of each orbit:
- 1st orbit (n=1): 2 electrons
- 2nd orbit (n=2): 8 electrons
- 3rd orbit (n=3): 18 electrons
- 4th orbit (n=4): 32 electrons
- 5th orbit (n=5): 50 electrons

Now, let's fill these orbits:
- 1st orbit: 2 electrons
- 2nd orbit: 8 electrons (Total = 10)
- 3rd orbit: 18 electrons (Total = 28)
- 4th orbit: 32 electrons (Total = 60)
- 5th orbit: 40 electrons (Total = 100)

Since we have filled 100 electrons, the outermost orbit is the 5th orbit (n=5).

### Step 3: Use the Formula for Orbital Radius
According to Bohr's model, the radius of the nth orbit is given by the formula:
\[ R_n = \frac{a_0 n^2}{Z} \]
where:
- \( a_0 \) = 0.53 Å (angstrom) = \( 53 \times 10^{-12} \) m
- \( n \) = 5 (the outermost orbit)
- \( Z \) = 100 (the atomic number)

### Step 4: Substitute Values into the Formula
Now we can substitute the values into the formula:
\[ R_5 = \frac{53 \times 10^{-12} \times 5^2}{100} \]

Calculating \( 5^2 \):
\[ 5^2 = 25 \]

Now substituting:
\[ R_5 = \frac{53 \times 10^{-12} \times 25}{100} \]

### Step 5: Calculate the Radius
Now, we can simplify:
\[ R_5 = \frac{1325 \times 10^{-12}}{100} \]
\[ R_5 = 13.25 \times 10^{-12} \text{ m} \]

### Step 6: Convert to Standard Form
Finally, we can express this in standard form:
\[ R_5 = 1.325 \times 10^{-11} \text{ m} \]

### Conclusion
The orbital radius of the atom `._100X^256` is approximately \( 1.325 \times 10^{-11} \) meters.

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